1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

132 Chapter 1 Fourier Series and Integrals

66.In analogy to Lemma 2 of Section 7, prove that

N∑− 1

n= 0

sin

((

n+

1

2

)

y

)

=

sin^2 (^12 Ny)

sin(^12 y).

67.Following the lines of Section 7, show that

σN(x)−f(x)= 2 N^1 π

∫π

−π

[

f(x+y)−f(x)

](sin(^12 Ny)

sin(^12 y)

) 2

dy.

This equality is the key to the proof of uniform convergence mentioned

in Exercise 65.

68.In a study of river freezing, E.P. Foltyn and H.T. Shen [St. Lawrence River

freeze-up forecast,Journal of Waterway, Port, Coastal and Ocean Engi-

neering, 112 (1986): 467–481] use data spanning 33 years to find this

Fourier series representation of the air temperature in Massena, NY:

T(t)=a 0 +

∑∞

n= 1

ancos( 2 nπt)+bnsin( 2 nπt).

HereTis temperature in◦C,tis time in years, and the origin is Oct. 1.

The first coefficients were found to be

a 0 = 6. 638 , a 1 = 5. 870 , b 1 =− 13. 094 , a 2 = 0. 166 , b 2 = 0. 583 ,

and the remaining coefficients were all less than 0.3 in absolute value.

The authors decided to exclude all the terms froma 2 andb 2 up, so their

approximation could be written

T(t)∼=a 0 +Asin( 2 πt+θ).

a. Find the average temperature in Massena.

b. FindA, the amplitude of the annual variation, and the phase angleθ.

c.Find the approximate date when the minimum temperature occurs.

d. Find the dates when the approximate temperature passes through 0.

e.Discuss the effect on the answer to partdif the next two terms of the

series were included.

69.In each part that follows, a function is equated to its Fourier series as

justified by the Theorem of Section 3. By evaluating both sides of the

equality at an appropriate value ofx, derive the second equality.